Chapter 2932 Review

1
Chapter 29-32 Review

• PHYS 2326-22

2
Concepts to KnowChapter 29

• Permanent Magnets
• Magnetic Poles
• N-Pole (north seeking)
• S-Pole (south seeking)
• Magnetic Field

3
Concepts to Know

• Magnetic field lines
• Magnetic Monopole (never found)
• Magnetic force on moving charge
• Cyclotron frequency
• Velocity selector
• Mass spectrometer

4
Magnetic Field

• Vector B represents the magnetic field with a
  direction from N-pole to S-pole
• Unit is Tesla
• alternate unit is Gauss 10-4 typical earth
  field 0.5 Gauss
• North pole of magnets attracted earths north
  pole which is opposite polarity

5
Units

• 1 Tesla 1 N/(Coulomb meter/s)
• a Coulomb/second is an ampere so
• 1 T 1 N/(A m)
• 104 Gauss (G) 1 Tesla (T)

6
Magnetic Force on Moving Charge
7
Electromagnetic Force

• on a moving charge

8
Electric Magnetic Force Differences

• Force for electric field is along the direction
  of the field line, force for magnetic field is
  perpendicular to the field
• Electric field force operates on any charge,
  Magnetic field force acts only on moving charge
• Electric field performs work when displacing
  charged particle, Magnetic field moves charges
  perpendicular to displacement performing no work.

9
Right Hand Rule
10
Right Hand Rule

• Cross product reverses sign when reverse vectors
  v x B -B x v
• Result is perpendicular to the plane that
  contains v and B.
• Direction for result F is the same as a screw
  moves when rotated in the direction from v to B
  (clockwise moves screw in and CCW moves screw out

11
Charged Particle in Uniform Field

• Magnetic force F is perpendicular to v and B
• Has constant magnitude qvB.
• Motion is in a circle for uniform B and no other
  forces acting on the moving charged particle
• centripetal acceleration mv2/r
• r mv/qB (top is momentum) eqn 29.3
• ? v/r qB/m angular speed
• T2pr/v 2 p/ ? 2 pm/ qB period

12
Cyclotron Frequency

• Cyclotron frequency is ? the angular speed
• A cyclotron is an essentially obsolete particle
  accelerator mostly now used in nuclear
  medicine.
• If there is a velocity component parallel to B
  there is no effect on the particle due to this
  component and the path becomes a helix

13
Velocity Selector

• An apparatus where there is an electric field
  perpendicular to a magnetic field and both
  perpendicular to the initial motion of a stream
  of charged particles one can create a velocity
  selector.
• FB qvB (direction right hand rule)
• FE qE (direction E)
• FB - FE (- is opposite directions)
• qE qvB so that v E/B (magnitude)
• Particles with velocity v go through undeflected,
  others are diverted

14
Mass Spectrometer

• Separates ions according to mass / charge ratio
• Bainbridge mass spec. beam enters a velocity
  separator then through a perpendicular magnetic
  field B
• Ions move in semicircle striking the detector at
  a point P. Negative ions are deflected one way,
  positive the other.

15
Mass Spectrometer Equations

• Starting with 29.3 the radius eqn rmv/qBo for
  uniform and perpendicular motion and Bo field.
• m/q Bo r/v by rearranging
• substituting 29.7 the velocity selector equation,
  vE/B
• m/q r Bo B/E

16
Concepts to KnowChapter 30

• Earths Field
• Magnetic Declination / Inclination
• 10 deg E of N declination here, inclination is
  up / down from horizontal
• Magnetic flux
• Gausss law for magnetism

17
Concepts to Know

• Magnetic Field Of Moving Charge
• Permeability Constant
• Biot Savart Law
• Magnetic Field of a long straight wire
• Right Hand Rule

18
Biot-Savart Law

• dB perpendicular to ds and to r where ds points
  in the direction of current I and r is the vector
  from the current element to P
• dB magnitude inversely proportional to r2 and
  proportional to current and magnitude ds and to
  the sine of ?, the angle between ds and r

19
Permeability of Free Space

• The constant µo in Biot-Savart is called the
  permeability of free space.
• µo 4p 1.0E-7 T m /A Tesla meters /amp.
• µo 1.2566 E-6 Wb/A m where Wb is a weber or
  Tesla meter2 (magnetic flux)
• Note too that the speed of light c is related
• c2 1/(µo eo ) the permeability and
  permitivity of free space

20
Finding B

• Biot-Savart shows a differential dB rather than
  the total B at a point P
• To find B one must integrate over the current
  distribution
• Note that the integrand is a vector

21
Magnetic field of Long Straight Wire

• See and go through Example 30.1
• Include eqn 30.5 on your next note card

P
?2
?1
a
I
O
x
22
Current Right Hand Rule

• Grasp wire with thumb pointed in current flow
  direction.
• Fingers are wrapped around in direction of B
  which tends to be circular and with a magnitude
  at radius a from the wire given by eqn 30.5

23
Field of Circular Coil

• Given a current loop of radius a at a distance x
  from the loop on the center axis find the B field
• by symmetry B will along the x axis

dB-
dB
ds
I
r
a
x
x
P
dBx
24
Circular Coil

• See Example 30.3
• Also note for a solenoid of multiple turns N that
  these multiply by N since that multiples I

25
Magnetic FluxGausss Law in Magnetism

• Electric flux is similar definition to magnetic
  field flux
• There is no magnetic monopole so no enclosed
  magnetic charge can be inside

26
Magnetic Field of a Moving Charge

• A moving charge creates a magnetic field in the
  same fashion a current element creates a charge
  except that differential current element is part
  of a whole while a moving charge is the whole
  contribution

27
Biot Savart Compared to Moving Particle

• Note the similarity between Biot Savarts current
  law and this moving particle
• I is charge per time moving past a point,
  velocity is distance per time
• For a point charge, dB becomes the whole field B

28
Example 1

• Two protons moving together along the x-axis
  separated by 1mm in the y axis with a velocity
  v1.00 E7 m/s
• a) find the electric force between them
• b) find the magnetic force between them
• c) find the total force
• d) what happens if v 100 m/s

29
Example 1

• Electric field Fe qE
• E k q/r2 1.439 E-3 N/C (a)
• Fe 2.3E-22 N repulsive
• B µo /4p q v/r2 1.6E-13
• Fb qvB 1.6 E-13 N/Am attractive (b)
• F Fe Fb 2.3E-22 N repulsive (c)
• Ratio Fe/Fb 899

30
Example 1

• If velocity reduces, E field unchanged, B field
  changes, Fb changes
• B 1.6E-18 N/Am
• Fb 2.56E-34 N attractive
• F 2.3E-22N repulsive
• R Fe/Fb 8.99 E11

31
Example 2

• Given a rectangular loop of 10cm x 20cm (a by b)
  with counter clockwise current I2A what is the
  magnitude and direction of magnetic field at the
  center point P
• B direction right hand rule out of page
• B BabBbcBcdBda 2Bab 2Bbc

a
d
I
P
b
c
32
Example 2
33
Concepts to KnowChapter 31

• Magnetic Induction
• Induced Current
• Induced EMF
• Faradays law
• Lenzs Law
• Motional Electromotive Force
• Induced Electric Fields
• Eddy Currents
• Maxwells Equations
• Superconductivity (Chapter 27.5)

34
Faradays Law

• Faradays Law of Induction
• An induced EMF is produced by a changing magnetic
  field.

35
Faradays Law

• Assuming a uniform field B at an angle ? to the
  vector area A produces
• EMF -d (BA cos ?)/dt
• Magnitude of B can change with time
• Area enclosed by loop can change with time
• Angle ? between B and A can change with time
• Any combination of the above

36
Motional EMF

• Given conductor of length l moving at velocity v
  in magnetic field B
• Electrons in conductor experience magnetic force
  Fb downward creating a negative charge at the
  bottom and a positive charge at the top and
  creating a downward electric field E which
  creates an upward force on electrons Fe

x
x
x
Fe
x
x
x
l
E
Fb
V
x
x
x
37
Motional EMF

• Charges accumulate until a balance is reached
  where the upward force Fe qE balances with the
  downward magnetic force on the electron Fb qvB
  or E vB
• Using the voltage relation ?V El for
  equilibrium becomes ?V El Blv
• ?V is potential but E is not electrostatic
• This En is an induced EMF
• It is maintained as long as the motion continues

38
Motional EMF

• The more general case for the induced EMF when v
  B and l are not mutually perpendicular

39
Lenzs Law

• The induced current in a loop is in the direction
  that creates a magnetic field that opposes the
  change in magnetic flux through the area enclosed
  by the loop

40
Induced EMF and Electric Fields

• An electric field is created in the conductor as
  a result of the changing magnetic flux
• Independence of test charges suggests that a
  changing magnetic field generates an electric
  field in empty space even without a conducting
  loop
• This is a nonconservative force

41
Eddy Currents

• According to Lenzs law currents are induced by a
  changing magnetic flux.
• Eddy currents are induced in bulk pieces of metal
  moving through a magnetic field.
• These are in the direction to create a magnetic
  field to oppose the changes in the existing
  magnetic field.

42
Maxwells Equations

• See chapter 34.2
• The basis of all electromagnetic phenomenon

43
Example 1

• Given a coil of wire with 100 turns and a radius
  of 5cm and a uniform magnetic field perpendicular
  to the coil plane, an increasing field B with
  rate 0.2T/s and a resistance of 50 0hms, find a)
  the emf induced, d) the direction of the magnetic
  field created by the coil, e) the power required
  to keep the original field increasing by 0.2 T/s
  in the z direction

44
Example 1

• r0.05m, N100, B (0.2T/s)t, R 50 O
• emf -N dFB /dt, FB is magnetic flux, A pr2
• A p(0.05)2 0.00785 m2
• B is some unknown value at time t0 but it
  changes at 0.2 T/s so assume its 0 at t0
• FBBA (0.2 t) (0.00785) 0.00157t Tm2
• emf -(100)(0.00157) -0.157V

x
x
x
x
x
x
r
x
x
x
45
Example 1

• b) Current? emf IR, Iemf/R
• I -0.157V/50 -0.00314A -3.14mA
• c) Direction of current? by Lenzs law opposes
  change of B which is increasing in the z
  direction. Therefore the current is flowing to
  create an opposing B field so by right hand rule,
  Bc is z and current is CCW.
• d) See above
• e)P I emf I2 R 0.493mW

46
Example 2

• 2 parallel long straight wires separated by
  L30cm are connected by on the left end and a
  sliding bar placed across the two wires and slid
  to the right at v2m/s, This is in the presence
  of the uniform field B10T directed into the
  paper. The bar has 10 Ohms resistance, the rest
  none. Find a) emf induced, b) current in the
  bar, c)magnetic force on the bar, d)power
  required to pull the bar, e) power dissipated in
  the bar

47
Example 2

• L 0.3m, xvt A Lx, as a f(t) Lvt
• a) emf dFB /dt, FB BA for uniform field
• FB BLvt so emf vBL (2)(10)(.3) 6V
• Flux into the page is increasing because of A.
  The current generated FB must be out of the page
  by Lenzs law so by RHR current is up.
• b) emf IR , E6/10 0.6 A

48
Example 2

• c) eqn 29.10 F ILxB force on a moving current
  carrying wire
• F0.6 0.3 10 1.8N opposing motion (left)
• d)power required, eqn 31.7 PF v
• P (1.8)(2) 3.6 W
• e) dissipation P I2R (0.6)2 10 3.6W

49
Concepts to KnowChapter 32

• Self Induction
• Mutual Inductance
• Inductors
• Magnetic Field Energy
• RL Circuit
• LC Circuit
• LRC Circuit

50
Self Induction

• Must distinguish between emf from sources like a
  battery and induced emf from changing magnetic
  fields

51
Mutual Inductance

• Magnetic flux variation in one circuit can cause
  a magnetic flux variation in another circuit.
  Note that this can be by intention or by
  accident.
• By intention, one can have a transformer
• By accident, one relay might cause another relay
  to close or open. or noise to be injected in one
  circuit by another

52
Mutual Inductance

• emf2 -N2 dF12/dt -N2 d(M12 I1/N2)/dt
• -M12 dI1/dt , so emf1 -M21 dI2/dt
• Note that also M12 M21 M
• That is a current in coil 1 generates current in
  coil 2 and a current in coil 2 generates a
  current in coil 1

53
Magnetic Field Energy

• Given an applied emf across an inductor in series
  with a resistance,

54
Transformer

• A transformer is two coils with total mutual
  induction Read Chapter 33.8

55
Permeability

• Amperes law applied to a toroid
• Note that the magnetic field B depends upon µo
  the permeability of free space. This is for the
  area in the torus that the winding goes around
• What happens if its not a vacuum?
• Most transformers and solenoids have metal cores
  in the coils

56
Permeability

• Note this doesnt seem to even be in this text
  book
• In the same concept as with dielectrics for
  capacitors there is a magnetic equivalent to the
  dielectric constant Km B/Bo relative
  permeability
• Note though that K for dielectrics ranged from 1
  for a vacuum upwards to ??
• Km is 1 for a vacuum, gt 1 for paramagnetic
  materials, slightly smaller than 1 for
  diagmagnetic materials and gtgt 1 for ferromagnetic
  materials

57
Permeability

• µ µo Km is the permeability of a material

58
Example 1

• Tesla coil has N1 turns along length l of a
  hollow tube and a second coil of N2 what is a)
  the mutual inductance b) if N110,000 and N2
  100, l 1m, radius 1cm what is the value of M?
  c) if a radio frequency of 1000 KHz is sent
  through coil 2 so that current oscillates with
  amplitude of 100ma what is the average magnetic
  flux through coil1 d)max current through coil1
  e)max induced emf in coil1 f)back emf in coil 1?

59
Example 1

• M N2 FB2 /I1, FB2 FB1 B1A
• A pr2 3.14(0.01)2 0.000314 m2
• B1µo N1 I1/l substituting for B1
• M (µo N1 I1/l ) N2 A / I1 µo N2 N1 A I1 / l
• b) M 3.946 E -4 H
• c) max flux?
• I2 Iosin ?t , ? 2 pf (angular freq.)6.28E6
• rearranging M, FB2 FB1 M I2 / N2
• (3.946E-4)(0.1)/(100) 3.948E-7 Wb

60
Example 1

• d) coil 1 current I1 I2 N2/N1
  (0.1)(100)/(10,000) 1.0 E-3A
• e) emf1 -M dI2 /dt , I2 Io sin ?t
• dI2 /dt Io ? cos ?t
• emf1 -M Io ? cos ?t
• emf1max M Io ? (3.948E-4)(0.1)(6.28E6) 248
  V
• f) emf2max emf1max N2/N1 (248)(100/10,000)
  2.48 V

61
Example 2

• a) Inductance of a long solenoid length l and
  area A with N turns? b)if 2m long 2cm radius and
  2000 turns? c) if current decreased from 4A to 0
  in 2 microseconds what is magnitude and direction
  of the self induced emf ? d) what is the energy
  stored in the solenoid at the beginning of the 2
  microsecond interval? e) How much electrical
  power is dissipated during this time?

62
Example 2

• inductance
• Substituting for B in L

63

• b) Inductance value
• A 1.257E-3
• L (1.2566E-6)(2000)2(1.257E-3)/2.0
• 3.159E-3 H
• c) emf?
• emf (3.159E-3)(4-0)/(2.0E-6) 6318V
• in direction of current trying to stop field
  collapse by trying to maintain current

64

• d) Energy?
• U1 (1/2) (3.158E-3)(4)2 2.52E-2 Joules
• e) Power
• P (2.52E-2)/(2E-6) 12,632 W
• why so high? Its also about timing too

65
RL Circuits

• Review 32.2

66
LC Circuits

• Study Chapter 32.5
• Given a capacitor and an inductor at time t0
  with the capacitor being connected in series to
  the inductor with Qmax charge on it there will be
  an oscillation. If no resistance is there to
  dissipate the energy, it will continue to
  oscillate

67
RLC Circuit

• Study Chapter 32.6
• The difference between an LC and RLC circuit
  other than resistance exists in all circuits is
  that R is a dissipative element that absorbs
  energy as current flows